Completely multiplicative functions with sum zero over generalised prime systems

Abstract

AbstractCMO functions multiplicative functions f for which $$\sum _{n=1}^\infty f(n) =0$$ ∑ n = 1 ∞ f ( n ) = 0 . Such functions were first defined and studied by Kahane and Saïas [14]. We generalised these to Beurling prime systems with the aim to investigate the theory of the extended functions and we shall call them $$CMO_{\mathcal {P}}$$ C M O P functions. We give some properties and find examples of $$CMO_{\mathcal {P}}$$ C M O P functions. In particular, we explore how quickly the partial sum of these classes of functions tends to zero with different generalised prime systems. The findings of this paper may suggest that for all $$CMO_{\mathcal {P}}$$ C M O P functions f over $$\mathcal{{N}}$$ N with abscissa 1, we have $$\begin{aligned} \sum _{\tiny \begin{array}{c} n \le x \\ n\in \mathcal {N} \end{array}} f(n) = \Omega \Big (\frac{1}{\sqrt{x}} \Big ). \end{aligned}$$ ∑ n ≤ x n ∈ N f ( n ) = Ω ( 1 x ) .